Exact Root-Label Realization Loci in Low-Cost Proximity-Decorated Binary Quadratic Paths

Abstract

Let $q\in k[[x,y]]\otimes\Sym^2(V^*)$ be a primitive framed binary quadratic germ over an algebraically closed field of characteristic zero. A maximal primitive blowup path carries multiplicities, free/satellite proximity, and exceptional types $B$, $M$, and $N$, where a constant-root branch vertex $B$ also carries an exact repeated-root class on the rank-one Veronese conic $\mathcal{C}=\nu_2\bigl(\PP(V^*)\bigr)\subset\PP\bigl(\Sym^2(V^*)\bigr).$ A preceding path-local classification under the square-cost bound $\sum\mu_i^2\le 12$ produced a $124$-row proximity-decorated envelope, with $119$ realizable rows and $5$ projective-transport obstructions. We now restore all exact constant-root labels and classify, for every row, the constructible locus of labels realized by one primitive germ. After quotienting only by the equalities already forced by projective-class transport, every nonempty row has at most two independent root-label components. The exact profile is $\boxed{124=46\{\mathrm{pt}\}+45\mathcal{C}+21\Delta_C+7(\mathcal{C}^2\setminus\Delta_C)+5\varnothing .}$ The two-component part is sharp: $21$ rows realize only the diagonal orbit, $7$ only the ordered off-diagonal orbit, and no row realizes both. For free chains, the mechanisms are a Newton-support overlap forcing equality and a multiplicity-drop-one nonbranch pencil forcing inequality. For nonfree chains, the same degree-one anchor pencil $sP+tQ$ forces opposite conditions according as it is exposed at an interior $B_1$ vertex or at the terminal $N_1$ vertex. Explicit primitive nonzero-discriminant normal forms realize every allowed orbit. The classification is path-local; it does not assert sibling compatibility, whole-tree glueing, or a finite count of exact labelled assignments.